Question

set up a proportion to solve for x in the following similar triangles.

Explanation:

Step1: Use property of similar - triangles

For similar triangles, the ratios of corresponding sides are equal. The ratio of the side of length 24 in the first triangle to the side of length \(x - 2\) in the second triangle is equal to the ratio of the side of length 18 in the first triangle to the side of length 12 in the second triangle. So, \(\frac{24}{x - 2}=\frac{18}{12}\).

Step2: Cross - multiply

Cross - multiplying the proportion \(\frac{24}{x - 2}=\frac{18}{12}\) gives \(24\times12 = 18\times(x - 2)\).

Step3: Simplify the equation

First, calculate \(24\times12=288\) and \(18\times(x - 2)=18x-36\). So the equation becomes \(288 = 18x-36\).

Step4: Solve for \(x\)

Add 36 to both sides of the equation: \(288 + 36=18x\), which simplifies to \(324 = 18x\). Then divide both sides by 18: \(x=\frac{324}{18}=18\).

Answer:

The proportion to solve for \(x\) is \(\frac{24}{x - 2}=\frac{18}{12}\) and \(x = 18\).